ABSTRACT

In this paper, an aggregate homotopy method is proposed to solve unconstrained minimax problems. The homotopy mapping is constructed by the linear homotopy and the aggregate function for the objective function, and the homotopy parameter is also used as the smoothing parameter of the aggregate function. Under some general assumptions, the existence and global convergence of a smooth homotopy path are proved for almost all starting points in $R^n$ or a ball set, and a stationary point of the unconstrained minimax problem can be obtained. A path-following procedure is introduced to numerically trace the homotopy path. When the smoothing parameter becomes small enough, an alternative strategy of the path-following procedure is given to reduce the ill-conditioning of the aggregate function, it uses the Newton method to solve a nonlinear system, which is derived from the first order optimality conditions for the unconstrained minimax problem. Numerical results show that the proposed method is efficient and robust.

摘要

在本文中，提出了一种聚合同伦方法来解决无约束极大极小问题。同伦映射由线性同伦和聚合函数为目标函数构造，同伦参数也用作聚合函数的平滑参数。在一些一般假设下，几乎所有的起点都证明了光滑同伦路径的存在性和全局收敛性${\mathrm{电阻}}^n$或一个球集，可以得到无约束极大极小问题的一个驻点。引入路径跟踪程序以数值跟踪同伦路径。当平滑参数变得足够小时，给出了一种路径跟随过程的替代策略，以减少聚合函数的病态，它使用牛顿法求解非线性系统，该非线性系统由一阶最优条件导出对于无约束极大极小问题。数值结果表明，所提出的方法是有效和鲁棒的。